Yes. There are a few ways.
Insert and break link:
The first thing I would try is inserting one design into the other and then right click on the linked file and use Break Link.
Copy and Paste:
Within a single project components can be copied between designs. In this case, you need to right click on the top of the browser (the root), and pick Create Components from Bodies. But if your sketches and bodies start at the root, they will not paste into the other design.
To make this work you must follow the best practices for assembly modeling. Specifically, you must have from the start used this workflow:
1. Create Component (RMB on the root of the browser)
2. Activate component. See below.
3. Create all sketches and bodies inside this component.
Otherwise, what happens is Fusion will deliver pasted components without parametric information. You can still add features or modify them, but the sketches will be gone.

The other benefit, aside from Copy and Paste of parametric components, is that this kind of workflow allows you to filter the timeline for each component when you want to work on it. See below.

Phil Eichmiller
Software Engineer
Quality Assurance
Autodesk, Inc.